Taylor-Rule-Type Reaction Functions

• Taylor-rule-type reaction functions are formal models linking a bank’s policy rate to inflation and output gaps.
• They incorporate features like time variation, nonlinearity, and risk sensitivity to capture regime shifts and enhance forecasting.
• Estimation techniques such as OLS, Lasso, Bayesian methods, and machine learning enable robust structural analysis of monetary policy.

A Taylor-rule-type reaction function is a formal empirical or theoretical mapping between a central bank’s policy instrument (typically an overnight rate such as the federal funds rate) and key macroeconomic indicators, most commonly the inflation deviation from target and the output gap. These reaction functions serve as workhorse representations of monetary policy behavior in the literature, enabling estimation, forecasting, and structural analysis of central banks’ responses to changes in macroeconomic conditions.

1. Fundamental Specification and Evolution

The canonical Taylor rule specifies the policy rate as a linear function: $$i_t = r^* + \beta (\pi_t - \pi^*) + \gamma (y_t - y_t^*)$$ where $i_t$ is the nominal interest rate, $r^*$ the equilibrium real rate or intercept term, $\pi_t$ inflation, $\pi^*$ inflation target, $y_t$ actual output, $y_t^*$ potential output, and $\beta, \gamma$ are policy response coefficients. Most empirical studies and model implementations allow for generalizations encompassing lagged interest rates, asset prices, exogenous shifters, nonlinearities, and time-varying coefficients (Tehranian, 2023, Karakas, 2023).

Recent research extends the classical framework along three major axes:

• Time-variation: Parameters are permitted to evolve, capturing shifts in policy regimes, structural breaks, and state-dependent behavior (Callot et al., 2014, Byrne et al., 2014).
• Nonlinearity: Nonlinear or machine learning models capture complex dependencies and asymmetric reactions (Karakas, 2023).
• Risk-sensitivity: The objective function may maximize alternative criteria, such as conditional quantiles of the policy loss, mapping to observed hawkish/dovish attitudes (Montes-Rojas et al., 28 Oct 2025).

2. Estimation Methodologies

Taylor-rule-type reaction functions are estimated using several econometric and machine learning methodologies:

• Ordinary Least Squares (OLS): Yields empirical weights for inflation and output gaps, often revealing less weight on output gap than theoretical rules (Karakas, 2023, Tehranian, 2023).
• Lasso and Adaptive Lasso: Used in high-dimensional VARX systems to induce sparsity and detect time-varying or regime-switching behavior. Parsimoniously time-varying models assume parameter changes are regularly zero, allowing estimation of periods of constancy, abrupt change, or persistent drift (Callot et al., 2014). Objective function:

$$\min_{\bm{\alpha}_t} \sum_{t=1}^T (i_t - \alpha_{0,t} - \alpha_{1,t} \pi_t - \alpha_{2,t} y_t)^2 + \lambda \sum_{j=1}^p |\alpha_{j,t}|$$

• Bayesian/MCMC (Kalman Filter, Gibbs Sampling): Enables full probabilistic inference for state-space models with time-varying parameters and measurement equations (Byrne et al., 2014). TVP dynamics:

$\phi_{jt} = \phi_{j,t-1} + \eta_{j,t}$

• Machine Learning (Feedforward Neural Network): Nonlinear mapping from macro variables to policy rate, leveraging gradient descent for error minimization and improved out-of-sample fit (Karakas, 2023).
• Indirect Inference of Risk Preferences: Mapping observed policy rate to the quantile index $\tau$ that best matches the quantile-utility optimal policy (Montes-Rojas et al., 28 Oct 2025).

3. Structural Instability and Time-Varying Behavior

Empirical results across the literature robustly document substantial instability in the estimated coefficients of Taylor-rule-type reaction functions, especially around major economic shocks (e.g., Volcker disinflation, financial crises). TVP models with Bayesian estimation or Lasso-type penalization capture such changes, and simulation evidence shows pronounced breaks in policy response—revealed as sustained shifts or abrupt discontinuities in parameters (Callot et al., 2014, Byrne et al., 2014).

This motivates the use of parsimonious models, where the degree of parameter variation is endogenous. Interpreting the selection of nonzero increments reveals underlying regime switches and periods of policy inertia. In exchange rate forecasting, time-varying Taylor rules notably outperform constant-parameter models and random walk benchmarks particularly during periods of policy turmoil (Byrne et al., 2014).

4. Nonlinearity and Augmentation

Linear empirical Taylor rules systematically underperform in matching the actual path of policy rates, especially through periods of asset-price volatility and post-bubble recessions. Data-driven nonlinear estimation via neural networks yields rate predictions that almost perfectly match actual implemented rates except in situations not covered by simple Taylor-type inputs (e.g., bubble burst recessions) (Karakas, 2023). Although asset prices (e.g., stock market changes) have been considered as supplements, research shows that including them does not significantly improve explanatory power or robustness of the Taylor rule for US or UK policy over 1990–2020, as measured by t-statistics and adjusted $R^2$ (Tehranian, 2023).

5. Theoretical Extensions and Optimal Policy

Taylor-rule-type reaction functions may arise as reduced form solutions to dynamic programming problems. Recent work shows that if the central bank maximizes quantile utility rather than expected utility, the resulting Taylor rule is indexed by a quantile level $\tau$ which reflects the authority’s risk attitude. The generalized Bellman equation is: $$v(x, z) = \sup_{i \in \Gamma(x, z)} \left\{ u(x, i, z) + \beta Q_\tau [v(\phi(x, i, z'), z') | z] \right\}$$ yielding a policy rule with a "risk management" term depending on quantiles, not just expected values. This approach rigorously maps dovish/hawkish periods to the estimated $\tau$ index and demonstrates that observed Fed behavior is usually dovish except during episodes of aggressive tightening (Montes-Rojas et al., 28 Oct 2025).

Within the New-Keynesian framework, the feedback mechanism embedded in Taylor rules critically affects model stability. Ramsey optimal policy (predetermined instrument, negative feedback) produces stability, whereas the standard Taylor principle (forward-looking instrument, positive feedback) can yield a Hopf bifurcation and dynamic instability. The correct anchoring of expectations and feedback specification is essential for well-posed model dynamics (Chatelain et al., 2020).

6. Policy, Fiscal Coordination, and Price-Level Implications

In continuous-time stochastic general equilibrium settings, Taylor-rule-type reaction functions for the policy rate can be rigorously embedded alongside stochastic fiscal policy and explicit modeling of shocks (Brownian motions, default terms). The result is a uniquely determined equilibrium, supporting the fiscal theory of the price level (FTPL), with no overdetermination or anomalies. The model demonstrates that price level determinacy is maintained when monetary policy follows a Taylor rule and fiscal policy independently reacts to output shocks, refuting critiques based on determinacy failures in discrete or deterministic models (Kofnov, 3 Mar 2024).

7. Summary of Empirical Coefficients and Specification

A representative table from the literature:

Model	Infl. ($\pi_t$)	Infl. Gap ($\pi_t-\pi^*$)	Output Gap ($y_t-y_t^*$)	Asset Price ($\delta$)	Adj. $R^2$
Taylor Rule (1993)	1	0.5	0.5	—	—
OLS (Karakas) (Karakas, 2023)	0.705	0.525	0.13	—	—
US Taylor (Tehranian, 2023)	—	0.91	0.45	0.012 (ns)	0.30
UK Taylor (Tehranian, 2023)	—	1.26	0.51	0.025 (ns)	0.28

References to Key Papers

• Time-varying and parsimonious parameter estimation: (Callot et al., 2014, Byrne et al., 2014)
• Nonlinear and machine learning approaches: (Karakas, 2023)
• Asset price augmentation and empirical stability: (Tehranian, 2023)
• Quantile utility optimal policy: (Montes-Rojas et al., 28 Oct 2025)
• Ramsey vs. Taylor rule bifurcation: (Chatelain et al., 2020)
• Continuous-time stochastic equilibrium and FTPL: (Kofnov, 3 Mar 2024)

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In sum, Taylor-rule-type reaction functions constitute a flexible, empirically and theoretically grounded class of models central to the analysis of monetary policy. Their specification, estimation, and generalization—particularly along the axes of time variation, nonlinearity, risk sensitivity, and economic stability—define the evolution of contemporary macroeconomic modeling and policy evaluation.